2.1.13 Suppose that \"Ford is tired and Zaphod has two heads\" is a false statem

Question

2.1.13 Suppose that "Ford is tired and Zaphod has two heads" is a false statement. What can we conclude if we discover each of the following? Theat the two scenarios separately. (a) Ford is tired. (b) Ford is tired if and only if Zaphod has two heads. (a) Do there exists propositions P, Q such that both P (b) Do there exist propositions P, Q such that both P Q and its converse are false? Justify your answers by giving an example or a proof that no such examples exist. 2.1.14 Q and its converse are true? 2.1.15 Let R be the proposition "The summit of Mount Everest is underwater". Suppose that S is a proposition such that (RVS)(RAS) is false. (a) What (b) What if, instead, (R V S)-(R S) is true? Hopefully it is obvious to you that R is false. can you say about S? 2.1.16 (Hard)Suppose that P.Q are propositions. Argue that any of the 16 possible truth tables T F T/F F T T/F F F T/F represents an expression ? created using only P and Q and the operations A,v,-. Can you extend your argument to show that any truth table with any number of inputs represents some logical expression? 20

Explanation / Answer

2.1.13

since "Ford is tired and Zaphold has two heads" is a false statement , this means either one of them is true or both are false statement as there is "and" operator between them .

a) Ford is tired = This statement will be true considering that "Zaphold have two heads" is a false statement

or this is a false statement itself

b) Ford is tired if and only if Zaphold has two heads = This statement is false because if Zaphold has two heads then ford will be tire since one of them is surely false this statement is false

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